1. Bullet motion

First we break down the initial velocity into its horizontal and vertical components:

We know that the acceleration in the horizontal component is 0, it is a constant velocity. Meanwhile in the vertical component we have an acceleration of :

Kinematic equations for projectile motion

Horizontal motion

The rate of change of horizontal position is constant (it changes every ):

The velocity of horizontal position doesn't change, which means it's constant.
AKA the rate of change of the rate of change of the horizontal position is 0(velocity never changes):

The actual formula for x axis:

Hint

This is just the formula for Non-Accelerated Linear Motion.

Vertical motion

The vertical position changes space units every plus the gravitational pull.
The rate of change of vertical position is not constant because the gravitational pull gets bigger over time:

The rate of change of vertical position also changes because of acceleration, it changes every :

The actual formula for axis:

Hint

This is just the formula for Accelerated Linear Motion, where the acceleration is gravity.

Trajectory equation

The following equation tells us at the object is at position (time is no more).

How does one derive this?

Horizontal motion equation

Vertical motion equation

Solving the horizontal motion equation for time

Now we can plug this into the vertical motion equation, basically its like putting in the formula the time at a certain .

Substituting into the vertical motion equation

Expanding and simplifying the equation

Substituting and with and

Caution

Isn't it just better to compute the fucking time and plug it in directly? instead of using this shit giant formula?

Horizontal reach

How do we derive this formula?

This is just the velocity multiplied by time to obtain how much we traveled in that time.
The time is the time of arrival(or time at which the object reaches the ground), which we compute as:

Maximum height

Exercise